Apart from this "superior tone" that the intuitionist may adopt (to use a Kantian expression designating Jacobian realism, which claims that we have an intuition of the thing in itself), this doctrine can be based on an interesting argument, originally formulated by moral intuitionism:
1/ No moral statement can be derived from purely non-moral statements
2/ Hence: for a moral statement to be derived from other assertions, these assertions must include moral statements
3/ Hence: for a moral belief to be inferentially justified, its justification must invoke other moral beliefs.
4/ However, to avoid infinite regression or circularity, the process of inferential justification must have an end
5/ Hence: there must be some self-evident moral beliefs ¹

This could be transposed to the axiological field as follows:
1/ A value judgement can only be justified on the basis of another value judgement (or axiological belief)
2/ However, to avoid infinite regression or circularity, the process of inferential justification must have an end
3/ Hence: there must be some axiological beliefs that are self-evident.

Now we see why axiological intuitionism is an attractive doctrine. First of all, intuitionism makes a problem disappear as such, thus relieving the human mind of a burden it felt incapable of lifting. Secondly, it gives everyone the extraordinary power to know what is valuable and what is not, a divining power at least as valuable as that of being able to predict tomorrow's weather.
Please note that I take 'axiological intuitionism' in a very broad sense, including not only doctrines that support the existence in us of a faculty - intuition - that immediately grasps values, but also theories of values that are based simply on the obvious, i.e. on a value judgement recognised as self-evident. This leads me to call ‘intuitionist’ thinkers who have never presented themselves as such.

It now seems appropriate to ask the intuitionist what his intuition tells him about the value of things, i.e. what has value, or what is the universal hierarchy?

1. Dictionnaire d'éthique et de philosophie morale, article « Moral realism »